Yes, it is possible to collaborate.  

I am interested in self-maps on metric spaces that have fixed points (generalizations of Contraction Mapping Principle mostly). What kind of problerms interest you?

Dear professor Jakub Jasinski,

I am working on "Geometry properties of Banach spaces such as uniformly rotund,

uniformly smooth and locally uniformly rotund, etc. play a vital

role in metric fixed point theory"

A Banach space

$(X,\|.\|)$ is said to be uniformly convex or

uniformly rotund (UR) if for all $y_n, x_n \in X$ satisfying

$\displaystyle\lim_{n\to

\infty}(2\|y_n\|^2+2\|x_n\|^2-\|y_n+x_n\|^2)=0$

we have $\displaystyle\lim_{n\to \infty}\|y_n-x_n\|=0,$

and a Banach space $X$ is said to be locally uniformly rotund (LUR) if for

all $x, x_n \in X$ satisfying

$\displaystyle\lim_{n\to\infty}(2\|x\|^2+2\|x_n\|^2-\|x+x_n\|^2)=0$

we have $\displaystyle\lim_{n\to \infty}\|x-x_n\|=0.$

The uniformly rotundity and uniformly smoothness are important

geometric conditions on a Banach space, since geometric conditions

such as uniformly rotundity and uniformly smoothness are sufficient

to imply the fixed point property.

so I am interested in geometry of Banach spaces that imply fixed point property

"Let $(X, \|.\|)$ be a Banach space and $C$ be a convex closed bounded subset of $X$.

A mapping $T: C \to C$ is called nonexpansive if for any $x, y \in C$

we have that $\|Tx - Ty\|\leq\|x - y\|$. $X$ is said to have the

fixed point property (abbr. FPP) if every nonexpansive mapping

defined from a closed convex bounded subset into itself has a

fixed point"

Any advice and suggestions will be highly appreciated.

thanks.

Dear professor JAbdalla Ahmad Tallafha ,

I am working on "Geometry properties of Banach spaces such as uniformly rotund,

uniformly smooth and locally uniformly rotund, etc. play a vital

role in metric fixed point theory"

A Banach space

$(X,\|.\|)$ is said to be uniformly convex or

uniformly rotund (UR) if for all $y_n, x_n \in X$ satisfying

$\displaystyle\lim_{n\to

\infty}(2\|y_n\|^2+2\|x_n\|^2-\|y_n+x_n\|^2)=0$

we have $\displaystyle\lim_{n\to \infty}\|y_n-x_n\|=0,$

and a Banach space $X$ is said to be locally uniformly rotund (LUR) if for

all $x, x_n \in X$ satisfying

$\displaystyle\lim_{n\to\infty}(2\|x\|^2+2\|x_n\|^2-\|x+x_n\|^2)=0$

we have $\displaystyle\lim_{n\to \infty}\|x-x_n\|=0.$

The uniformly rotundity and uniformly smoothness are important

geometric conditions on a Banach space, since geometric conditions

such as uniformly rotundity and uniformly smoothness are sufficient

to imply the fixed point property.

so I am interested in geometry of Banach spaces that imply fixed point property

"Let $(X, \|.\|)$ be a Banach space and $C$ be a convex closed bounded subset of $X$.

A mapping $T: C \to C$ is called nonexpansive if for any $x, y \in C$

we have that $\|Tx - Ty\|\leq\|x - y\|$. $X$ is said to have the

fixed point property (abbr. FPP) if every nonexpansive mapping

defined from a closed convex bounded subset into itself has a

fixed point"

Any advice and suggestions will be highly appreciated.

thanks.

@Dear prof,Abdalla Ahmad Tallafha,

How can we collaborate together?

Dear  Arash Irani,

There is the theory of holomorphic functions with domain and range contained in a complex Banach space.  Perhaps the most basic is the Earle-Hamilton fixed point theorem, which may

be viewed as a holomorphic formulation of Banach's contraction mapping

theorem.

[Earle-Hamilton theorem]

Let $G$ be a nonempty domain in a complex

Banach space $X$

and let $h:G \rightarrow G$ be a bounded holomorphic function. If

$h(G)$ lies strictly inside

$G$, then $h$ has a unique fixed point in $G$.

For example,  Cartheodory  distances   are  nonexpansive.

Let $G$ be bounded connected open subset of complex Banach space,

$p\in G$ and $v\in T_p G$. We define $k_G(p,v)= \inf\{ |h|$,

where infimum is taking over all $h\in T_0 \mathbb{C}$ for which

there exists a holomorphic function such that $f: \mathbb{U}

\rightarrow G $ such that $f(0)=p$ and $df(h)=v$.

One can prove

\begin{thm}

Suppose that $G$ and $G_1$ are bounded connected open subset

of complex Banach space and $f: G \rightarrow G_1$ is

holomorphic. Then $k_G(fz,fz_1)\leq k_G(z,z_1)$ for all

$z,z_1\in G$.

\end{thm}

\begin{thm}

Suppose that $G$ is bounded connected open subset of complex

Banach space and $f: G \rightarrow G$ is holomorphic, $s_0=

dist(f(G),G^c)$, $d_0=diam(G)$ and $q_0=

\frac{d_0}{d_0+s_0}$. Then $k_G(fz,fz_1)\leq q_0 k_G(z,z_1)$.

\end{thm}

best, MM

@Dear Dear prof,Miodrag Mateljević

Thank you for your co-operations

with my regards for you

Hello Dear 

i am new in this field as i have recently earned my M Phil degree in this field. i have worked in fuzzy metric space. i do want to work with you please send me some meterial to discuss with and share. we will work together. 

here is my e-mail: [email protected]

Muhammad Bilal

Dear Arash Irani, this my e- mail to collaborate, [email protected]

Yes I also want to do work in this direction.

@Dear Shabir Ahanger , thanks  for your message, I want know  about   your  experience in functional analysis, especially, Banach space theory, renorming theory and fixed point theory . because I think experience in these subjects is very important

with my regards for you

Dear  Arash Irani,

You may be interested in  

Banach contraction principle   for holomorphic function in Banach spaces. See   also Denjoy-Wolf f   theorem related to iterations of holomorphic function  and the project   on RG  related to Schwarz lemma , Caratheodory, Kobyashi metrics and applications in Complex Analysis  and the literature cited there.  See also the project related to fpt  and applications.

Bessaga paper  is a converse  and we can   ask  whether    various generalization  of 

the Banach contraction principle   are really essential.

Theorem(Denjoy-Wolf f ). Let D be the open unit disk in C and let f be a holomorphic function mapping D into D which is not an automorphism of D (i.e. a Möbius transformation). Then there is a unique point z in the closure of D such that the iterates of f tend to z uniformly on compact subsets of D. If z lies in D, it is the unique fixed point of f. The mapping f leaves invariant hyperbolic disks centered on z, if z lies in D, and disks tangent to the unit circle at z, if z lies on the boundary of D.  There is a version  for holomorphic functions on   unit ball in Banach space.

You may contact if still you are interested.

I am very interisting, fixed point and applications

My email is : [email protected].

You can also see my profile.

Best Regards,

Your,

H. A. Hammad